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Properties

Label 4-7440e2-1.1-c1e2-0-3
Degree $4$
Conductor $55353600$
Sign $1$
Analytic cond. $3529.39$
Root an. cond. $7.70770$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s + 6·7-s + 3·9-s − 4·11-s − 6·13-s − 4·15-s + 4·17-s − 4·19-s + 12·21-s + 8·23-s + 3·25-s + 4·27-s − 2·29-s + 2·31-s − 8·33-s − 12·35-s − 10·37-s − 12·39-s + 8·43-s − 6·45-s + 12·47-s + 16·49-s + 8·51-s + 4·53-s + 8·55-s − 8·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 2.26·7-s + 9-s − 1.20·11-s − 1.66·13-s − 1.03·15-s + 0.970·17-s − 0.917·19-s + 2.61·21-s + 1.66·23-s + 3/5·25-s + 0.769·27-s − 0.371·29-s + 0.359·31-s − 1.39·33-s − 2.02·35-s − 1.64·37-s − 1.92·39-s + 1.21·43-s − 0.894·45-s + 1.75·47-s + 16/7·49-s + 1.12·51-s + 0.549·53-s + 1.07·55-s − 1.05·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(55353600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(3529.39\)
Root analytic conductor: \(7.70770\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 55353600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.029356093\)
\(L(\frac12)\) \(\approx\) \(6.029356093\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
31$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.7.ag_u
11$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_o
13$D_{4}$ \( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_bg
17$D_{4}$ \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_ba
19$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_be
23$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_by
29$D_{4}$ \( 1 + 2 T + 56 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_ce
37$D_{4}$ \( 1 + 10 T + 96 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.37.k_ds
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.41.a_cs
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$D_{4}$ \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_c
59$D_{4}$ \( 1 - 14 T + 140 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_fk
61$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.61.a_o
67$D_{4}$ \( 1 - 10 T + 132 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.67.ak_fc
71$D_{4}$ \( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.71.as_ho
73$D_{4}$ \( 1 + 6 T + 80 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_dc
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$D_{4}$ \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.83.y_lm
89$D_{4}$ \( 1 - 18 T + 256 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_jw
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.980405364283979107675914284906, −7.80148087049724891677741592690, −7.46869545316400334061361100599, −7.31990502774759708007917562409, −6.79577909409954493849758227915, −6.74175470455095654461422939606, −5.68195352683126184586827553058, −5.37384113006513844910051860007, −5.22904831976120338092956437301, −4.89960103485956735914394297353, −4.40856502517852985122295944118, −4.28763397554037616274387884073, −3.75582259841375700237862137138, −3.28967903376864889575926587052, −2.84031488220301130184502262379, −2.46782801512251111762031584150, −1.94289685747915688300495631023, −1.93986148160625845292200798135, −0.873197683616142616393270016733, −0.67782093606722702832006427181, 0.67782093606722702832006427181, 0.873197683616142616393270016733, 1.93986148160625845292200798135, 1.94289685747915688300495631023, 2.46782801512251111762031584150, 2.84031488220301130184502262379, 3.28967903376864889575926587052, 3.75582259841375700237862137138, 4.28763397554037616274387884073, 4.40856502517852985122295944118, 4.89960103485956735914394297353, 5.22904831976120338092956437301, 5.37384113006513844910051860007, 5.68195352683126184586827553058, 6.74175470455095654461422939606, 6.79577909409954493849758227915, 7.31990502774759708007917562409, 7.46869545316400334061361100599, 7.80148087049724891677741592690, 7.980405364283979107675914284906

Graph of the $Z$-function along the critical line